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Total Nonnegativity of Infinite Hurwitz Matrices of Entire and Meromorphic Functions

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Abstract

In this paper we completely describe functions generating the infinite totally nonnegative Hurwitz matrices. In particular, we generalize the well-known result by Asner and Kemperman on the total nonnegativity of the Hurwitz matrices of real stable polynomials. An alternative criterion for entire functions to generate a Pólya frequency sequence is also obtained. The results are based on a connection between a factorization of totally nonnegative matrices of the Hurwitz type and the expansion of Stieltjes meromorphic functions into Stieltjes continued fractions (regular \(C\)-fractions with positive coefficients).

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Notes

  1. In general, the condition to be meromorphic is replaced by less restrictive \(F(\overline{z})=\overline{F(z)}\). Basic properties of \(\mathcal {R}\)-functions can be found, for example, in [14] and (for the meromorphic case) in [23]. In this paper we confine ourselves to meromorphic functions only.

  2. The notation \([a]\) stands for the integer part of \(a\).

  3. The same transformation also allows us to express the formulae (12)–(13) from [19, p. 121] as the relation (2.5).

  4. The separate convergence of numerators and denominators was shown by Śleszyński in [20].

  5. This fact was obtained by Maillet in [18]; see also [19, p. 150].

  6. In fact, even an estimate stronger than (2.17) is valid (cf. [20, p. 105]). For each tuple of distinct numbers \((i_1,i_2,\dots ,i_k)\) there is only one summand \(\Big (\sigma _{i_1}^{(j)}\sigma _{i_2}^{(j)}\cdots \sigma _{i_k}^{(j)}\Big )^{-1}\) in the right-hand side of (2.16). At the same time, the sum

    $$\begin{aligned} \sum _{i_1=1}^\infty \sum _{i_2=1}^\infty \dots \sum _{i_k=1}^\infty \Big (\sigma _{i_1}^{(j)} \sigma _{i_2}^{(j)} \cdots \sigma _{i_k}^{(j)}\Big )^{-1} \end{aligned}$$

    contains exactly \(k!\) such summands. Therefore,

    $$\begin{aligned} a_k^{(j)}< \frac{1}{k!} \left( a_1^{(j)}\right) ^{k} \text { for }k=2,3,4,\dots . \end{aligned}$$

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Acknowledgments

The author is grateful to Olga Holtz and Mikhail Tyaglov for helpful comments and stimulating discussions.

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  1. Sekretariat MA 4-2, TU-Berlin, Straße des 17. Juni 136, 10623 , Berlin, Germany

    Alexander Dyachenko

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Correspondence to Alexander Dyachenko.

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Communicated by Daniel Aron Alpay.

This work was financially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 259173.

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Dyachenko, A. Total Nonnegativity of Infinite Hurwitz Matrices of Entire and Meromorphic Functions. Complex Anal. Oper. Theory 8, 1097–1127 (2014). https://doi.org/10.1007/s11785-013-0344-0

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